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Introduction to Commutative Algebra

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Free Online Course: Introduction to Commutative Algebra provided by Swayam is a comprehensive online course, which lasts for 12 weeks long. The course is taught in English and is free of charge. Upon completion of the course, you can receive an e-certificate from Swayam.

IIT Madras NPTEL via Swayam
12 weeks long
Finished
Overview

  • This is an introductory course in Commutative Algebra where most basic tools on commutative rings and modules over commutative rings are developed. This course is essential for anyone who wants to do research in areas such as commutative algebra, algebraic geometry, algebraic number theory etc.
    INTENDED AUDIENCE : M.Sc. & Ph.D. Mathematics Students.PRE-REQUISITES : Linear Algebra, Basic group theory and Basic ring theory including ED, PID & UFD.

Syllabus

  • COURSE LAYOUT

    MODULE 1 : Rings, ring homomorphism, ideals, quotients, zero divisors, nilpotents and units.MODULE 2: Prime and maximal ideals, nilradical and Jacobsons radicalMODULE 3: Operations on ideals, extension and contraction.MODULE 4: Modules and module homomorphisms, Submodules and quotient modules, Operations on submodules, Direct sum and productMODULE 5: Finitely generated modules, Exact sequences, Tensor product of modules.MODULE 6: Restriction and extension of scalars, Exactness properties of the tensor product.MODULE 7: LocalizationMODULE 8: Integral dependence, Going-up and Going-down theorems.MODULE 9: Chain conditions, Noetherian ringsMODULE 10: Primary decomposition in Notherian rings.MODULE 11: Artinian rings

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